Problem: The quadratic $x^2 + 5x + c$ has roots in the form of $x = \frac{-5 \pm \sqrt{c}}{2}$. What is the value of $c$?
Answer: Using the quadratic formula $x = \frac{ - b \pm \sqrt {b^2 - 4ac} }{2a}$, we can find the roots of the quadratic. We find that $x = \frac{-5 \pm \sqrt{25-4c}}{2}$. Thus, setting our two expressions for $x$ equal to each other, we find that  \begin{align*}
\frac{-5 \pm \sqrt{25-4c}}{2} &= \frac{-5 \pm \sqrt{c}}{2} \quad \Rightarrow \\
25 - 4c &= c \quad \Rightarrow \\
c &= \boxed{5}.
\end{align*}